title

The Laplace Transform: A Generalized Fourier Transform

description

This video is about the Laplace Transform, a powerful generalization of the Fourier transform. It is one of the most important transformations in all of science and engineering.
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Error: @10:20, should be e^{-st}

detail

{'title': 'The Laplace Transform: A Generalized Fourier Transform', 'heatmap': [{'end': 475.726, 'start': 453.972, 'weight': 0.956}, {'end': 643.23, 'start': 616.116, 'weight': 0.713}], 'summary': 'Explores the laplace transform, a powerful tool simplifying solving systems and extensively used in control theory, compares it with the fourier transform, and discusses its applications in stabilizing functions and solving pdes and odes.', 'chapters': [{'end': 167.927, 'segs': [{'end': 107.419, 'src': 'embed', 'start': 66.232, 'weight': 0, 'content': [{'end': 76.38, 'text': 'You can take a system and subtract about two or three years of advanced math from how hard it is to solve that system just by applying the Laplace transform.', 'start': 66.232, 'duration': 10.148}, {'end': 83.285, 'text': 'So, for example, if you have a partial differential equation, a PDE, under certain circumstances, you can Laplace,', 'start': 76.98, 'duration': 6.305}, {'end': 87.928, 'text': 'transform it and turn it from a PDE into an ODE, which is much simpler.', 'start': 83.285, 'duration': 4.643}, {'end': 98.614, 'text': 'Similarly, you can take an ODE and under some conditions you can transform it with the Laplace, transform into an algebraic equation which,', 'start': 88.789, 'duration': 9.825}, {'end': 102.416, 'text': 'again this goes from college to high school kind of solution techniques.', 'start': 98.614, 'duration': 3.802}, {'end': 107.419, 'text': 'And the Laplace transform is also extremely useful in control theory.', 'start': 103.016, 'duration': 4.403}], 'summary': 'Laplace transform simplifies advanced math, making systems easier to solve, useful in control theory.', 'duration': 41.187, 'max_score': 66.232, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/7UvtU75NXTg/pics/7UvtU75NXTg66232.jpg'}, {'end': 188.481, 'src': 'embed', 'start': 150.177, 'weight': 4, 'content': [{'end': 152.038, 'text': 'and Laplace was truly one of the greatest.', 'start': 150.177, 'duration': 1.861}, {'end': 155.54, 'text': 'Fun fact about Laplace.', 'start': 153.018, 'duration': 2.522}, {'end': 163.764, 'text': "he was one of the first researchers ever to realize that when you're dealing with real-world data which has noise and isn't perfect,", 'start': 155.54, 'duration': 8.224}, {'end': 167.927, 'text': 'you have to look at that data through a probabilistic lens, through the lens of probability theory.', 'start': 163.764, 'duration': 4.163}, {'end': 174.691, 'text': 'For us, we take that for granted, but that was a huge deal back when Laplace lived in the second half of the 1700s and the early 1800s.', 'start': 168.607, 'duration': 6.084}, {'end': 175.291, 'text': "Okay, so let's jump in.", 'start': 174.751, 'duration': 0.54}, {'end': 188.481, 'text': 'The big idea here is that we know that we can Fourier transform nice, well-behaved functions that decay to zero at plus and minus infinity.', 'start': 178.593, 'duration': 9.888}], 'summary': 'Laplace pioneered probabilistic lens for real-world data in 1700s-1800s', 'duration': 38.304, 'max_score': 150.177, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/7UvtU75NXTg/pics/7UvtU75NXTg150177.jpg'}], 'start': 7.073, 'title': 'Laplace transform', 'summary': 'Discusses the laplace transform, a powerful tool in mathematics and engineering, simplifying solving systems, converting pdes to odes, and extensively used in control theory.', 'chapters': [{'end': 167.927, 'start': 7.073, 'title': 'Laplace transform: generalized fourier transform', 'summary': 'Discusses the laplace transform, a generalized version of the fourier transform, which simplifies solving systems, converting pdes to odes, and is extensively used in control theory, making it a powerful tool in mathematics and engineering.', 'duration': 160.854, 'highlights': ['The Laplace transform is a generalized version of the Fourier transform, simplifying the solution of systems and converting PDEs to ODEs. The Laplace transform generalizes the Fourier transform to a larger class of functions, simplifying the solution of systems and converting PDEs to ODEs.', 'Applying the Laplace transform can simplify solving systems by subtracting advanced math, making the process much simpler. Applying the Laplace transform can simplify solving systems by subtracting advanced math, making the process much simpler.', 'Under certain conditions, the Laplace transform can transform a PDE into an ODE, simplifying the solution techniques from college to high school level. Under certain conditions, the Laplace transform can transform a PDE into an ODE, simplifying the solution techniques from college to high school level.', 'The Laplace transform is extensively useful in control theory, making it a powerful tool in engineering and mathematics. The Laplace transform is extensively useful in control theory, making it a powerful tool in engineering and mathematics.', 'Laplace was one of the first researchers to realize the importance of looking at real-world data through a probabilistic lens. Laplace was one of the first researchers to realize the importance of looking at real-world data through a probabilistic lens.']}], 'duration': 160.854, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/7UvtU75NXTg/pics/7UvtU75NXTg7073.jpg', 'highlights': ['The Laplace transform is extensively useful in control theory, making it a powerful tool in engineering and mathematics.', 'Applying the Laplace transform can simplify solving systems by subtracting advanced math, making the process much simpler.', 'Under certain conditions, the Laplace transform can transform a PDE into an ODE, simplifying the solution techniques from college to high school level.', 'The Laplace transform is a generalized version of the Fourier transform, simplifying the solution of systems and converting PDEs to ODEs.', 'Laplace was one of the first researchers to realize the importance of looking at real-world data through a probabilistic lens.']}, {'end': 314.82, 'segs': [{'end': 214.402, 'src': 'embed', 'start': 168.607, 'weight': 1, 'content': [{'end': 174.691, 'text': 'For us, we take that for granted, but that was a huge deal back when Laplace lived in the second half of the 1700s and the early 1800s.', 'start': 168.607, 'duration': 6.084}, {'end': 175.291, 'text': "Okay, so let's jump in.", 'start': 174.751, 'duration': 0.54}, {'end': 188.481, 'text': 'The big idea here is that we know that we can Fourier transform nice, well-behaved functions that decay to zero at plus and minus infinity.', 'start': 178.593, 'duration': 9.888}, {'end': 190.143, 'text': "So I'm just going to draw an example of that.", 'start': 188.621, 'duration': 1.522}, {'end': 200.231, 'text': 'If I have this nice Gaussian that goes to zero as x or t goes to plus or minus infinity, I can Fourier transform.', 'start': 190.663, 'duration': 9.568}, {'end': 202.033, 'text': "So I'm going to say Fourier transform, check.", 'start': 200.251, 'duration': 1.782}, {'end': 203.314, 'text': 'We can do the Fourier transform.', 'start': 202.093, 'duration': 1.221}, {'end': 214.402, 'text': "less well-behaved functions, so there are nastier functions out there, and I'm gonna draw a couple of them right now, like e to the lambda t.", 'start': 204.714, 'duration': 9.688}], 'summary': "Laplace's work in the 1700s laid the foundation for fourier transforms, applicable to well-behaved functions.", 'duration': 45.795, 'max_score': 168.607, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/7UvtU75NXTg/pics/7UvtU75NXTg168607.jpg'}, {'end': 314.82, 'src': 'embed', 'start': 266.306, 'weight': 0, 'content': [{'end': 269.349, 'text': "So let's think about this trigonometric function.", 'start': 266.306, 'duration': 3.043}, {'end': 275.516, 'text': "Now, again, this doesn't decay to zero at plus and minus infinity, but there are tricks you can play.", 'start': 270.27, 'duration': 5.246}, {'end': 280.502, 'text': 'So the most common trick is to multiply this by a window function w.', 'start': 275.857, 'duration': 4.645}, {'end': 286.406, 'text': 'where basically W is 1 on some window and 0 everywhere else.', 'start': 282.063, 'duration': 4.343}, {'end': 290.908, 'text': 'So now if I multiply W by my sine function or cosine function, it does have this nice property.', 'start': 286.606, 'duration': 4.302}, {'end': 295.091, 'text': 'And then I can take the limit as this window becomes infinitely large.', 'start': 291.729, 'duration': 3.362}, {'end': 299.173, 'text': "That's one way you can Fourier transform these signals, but again, it's kind of a pain.", 'start': 295.311, 'duration': 3.862}, {'end': 308.539, 'text': "So what I'm going to show you is how the Laplace transform is basically a weighted, one-sided Fourier transform for these nasty functions.", 'start': 299.533, 'duration': 9.006}, {'end': 311.059, 'text': "OK, that's all I'm going to show you today.", 'start': 309.359, 'duration': 1.7}, {'end': 313.56, 'text': "And then we're going to use it later on.", 'start': 311.419, 'duration': 2.141}, {'end': 314.82, 'text': 'OK, good.', 'start': 314.46, 'duration': 0.36}], 'summary': 'Laplace transform is a weighted, one-sided fourier transform for signals, useful for transforming tricky functions.', 'duration': 48.514, 'max_score': 266.306, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/7UvtU75NXTg/pics/7UvtU75NXTg266306.jpg'}], 'start': 168.607, 'title': 'Laplace transform vs. fourier transform', 'summary': "Compares fourier transform and laplace transform, emphasizing fourier transform's capability with well-behaved functions and limitations with nastier functions, and introduces laplace transform as a weighted, one-sided fourier transform for these functions.", 'chapters': [{'end': 314.82, 'start': 168.607, 'title': 'Laplace transform: fourier transform comparison', 'summary': 'Discusses the comparison between fourier transform and laplace transform, highlighting the ability of fourier transform to handle well-behaved functions and the limitations when dealing with nastier functions, and introduces the laplace transform as a weighted, one-sided fourier transform for these functions.', 'duration': 146.213, 'highlights': ['Laplace transform as a weighted, one-sided Fourier transform for nastier functions, addressing the limitations of Fourier transform in handling these functions.', 'Explanation of the limitations of Fourier transform in dealing with nastier functions such as e to the lambda t and the Heaviside function.', 'Demonstration of the ability of Fourier transform to handle well-behaved functions that decay to zero at plus and minus infinity.', 'Introduction of the concept of using a window function to multiply with trigonometric functions for Fourier transformation, albeit with complexities.']}], 'duration': 146.213, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/7UvtU75NXTg/pics/7UvtU75NXTg168607.jpg', 'highlights': ['Laplace transform as a weighted, one-sided Fourier transform for nastier functions, addressing the limitations of Fourier transform', 'Demonstration of the ability of Fourier transform to handle well-behaved functions that decay to zero at plus and minus infinity', 'Explanation of the limitations of Fourier transform in dealing with nastier functions such as e to the lambda t and the Heaviside function', 'Introduction of the concept of using a window function to multiply with trigonometric functions for Fourier transformation, albeit with complexities']}, {'end': 498.082, 'segs': [{'end': 421.48, 'src': 'embed', 'start': 348.1, 'weight': 0, 'content': [{'end': 363.523, 'text': "that's an exponential function e to the minus gamma t, so that f of t e to the minus gamma t goes to 0, as t goes to positive infinity.", 'start': 348.1, 'duration': 15.423}, {'end': 366.164, 'text': 'So only as t goes to positive infinity.', 'start': 364.083, 'duration': 2.081}, {'end': 367.964, 'text': "So the first thing we're going to do,", 'start': 366.904, 'duration': 1.06}, {'end': 374.667, 'text': "the solution is we're going to take our function f that's badly behaved and we're going to multiply it by a decaying exponential function.", 'start': 367.964, 'duration': 6.703}, {'end': 383.951, 'text': "So we're going to multiply it by a decaying exponential so that when those multiply together, it will go to 0 as t goes to positive infinity.", 'start': 374.687, 'duration': 9.264}, {'end': 394.125, 'text': 'Now you might be thinking well, we solved this problem in the positive t infinity direction, but now my function might blow up at negative t infinity.', 'start': 384.938, 'duration': 9.187}, {'end': 395.226, 'text': 't equals negative infinity.', 'start': 394.125, 'duration': 1.101}, {'end': 404.092, 'text': "So we don't just multiply by e to the minus gamma t, we also multiply by our handy heavy side function h of t, okay?", 'start': 395.806, 'duration': 8.286}, {'end': 419.56, 'text': "So now what we have is we're gonna define this big function big F of t is going to equal little f times e to the minus gamma t times heavy side of t.", 'start': 404.592, 'duration': 14.968}, {'end': 421.48, 'text': "All right, that's what we're gonna do.", 'start': 419.56, 'duration': 1.92}], 'summary': 'Using a decaying exponential function, the solution ensures f(t) goes to 0 as t goes to positive infinity.', 'duration': 73.38, 'max_score': 348.1, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/7UvtU75NXTg/pics/7UvtU75NXTg348100.jpg'}, {'end': 475.726, 'src': 'heatmap', 'start': 443.124, 'weight': 1, 'content': [{'end': 448.948, 'text': 'And it equals f of t e to the minus gamma t for t greater than or equal to zero.', 'start': 443.124, 'duration': 5.824}, {'end': 453.631, 'text': 'Good So this is the whole thing.', 'start': 449.969, 'duration': 3.662}, {'end': 462.878, 'text': "We take our badly behaved function, we multiply it by a stable exponential and a Heaviside function, and now we're going to Fourier transform big F.", 'start': 453.972, 'duration': 8.906}, {'end': 467.281, 'text': 'So the Laplace transform of little f is the Fourier transform of big F.', 'start': 462.878, 'duration': 4.403}, {'end': 469.582, 'text': "Good And I'm going to write this down over here.", 'start': 467.901, 'duration': 1.681}, {'end': 470.523, 'text': "I'm actually going to box it.", 'start': 469.602, 'duration': 0.921}, {'end': 475.726, 'text': "So I'm going to write down my Laplace transform pair just like we write down our Fourier transform pair.", 'start': 470.543, 'duration': 5.183}], 'summary': 'Laplace transform of function f(t) is fourier transform of f', 'duration': 26.458, 'max_score': 443.124, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/7UvtU75NXTg/pics/7UvtU75NXTg443124.jpg'}], 'start': 316.26, 'title': "Laplace's method and transform", 'summary': "Explains laplace's method for asymptotic approximations and discusses stabilizing functions for fourier transformation, including the use of stable exponential and heaviside function.", 'chapters': [{'end': 395.226, 'start': 316.26, 'title': "Laplace's method", 'summary': "Explains laplace's method for finding asymptotic approximations, where the solution involves multiplying a function by a decaying exponential function to make it approach zero as t goes to positive infinity.", 'duration': 78.966, 'highlights': ['The solution involves multiplying a function by a decaying exponential function so that f(t) * e^(-gamma*t) goes to 0 as t goes to positive infinity.', 'This approach is used to make the badly behaved function approach zero as t goes to positive infinity.']}, {'end': 498.082, 'start': 395.806, 'title': 'Laplace transform: stabilizing functions for fourier transformation', 'summary': 'Discusses the use of stable exponential and heaviside function to stabilize a badly behaved function, allowing its fourier transformation, with the laplace transform of little f equaling the fourier transform of big f.', 'duration': 102.276, 'highlights': ['The use of stable exponential and Heaviside function to stabilize a badly behaved function, allowing its Fourier transformation.', 'Defining big function big F of t as little f times e to the minus gamma t times heavy side of t.', 'The Laplace transform of little f is the Fourier transform of big F.']}], 'duration': 181.822, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/7UvtU75NXTg/pics/7UvtU75NXTg316260.jpg', 'highlights': ['The solution involves multiplying a function by a decaying exponential function so that f(t) * e^(-gamma*t) goes to 0 as t goes to positive infinity.', 'The use of stable exponential and Heaviside function to stabilize a badly behaved function, allowing its Fourier transformation.', 'Defining big function big F of t as little f times e to the minus gamma t times heavy side of t.', 'The Laplace transform of little f is the Fourier transform of big F.', 'This approach is used to make the badly behaved function approach zero as t goes to positive infinity.']}, {'end': 986.415, 'segs': [{'end': 525.383, 'src': 'embed', 'start': 499.242, 'weight': 2, 'content': [{'end': 510.367, 'text': "So the Fourier transform of big F is going to be we're gonna call that big F hat and it's gonna be a function of omega,", 'start': 499.242, 'duration': 11.125}, {'end': 521.062, 'text': "and that's going to equal the integral from minus infinity to infinity, of big F, of t e to the minus i, omega t dt.", 'start': 510.367, 'duration': 10.695}, {'end': 523.222, 'text': 'This is what we always do when we Fourier transform.', 'start': 521.102, 'duration': 2.12}, {'end': 525.383, 'text': "That's the Fourier transform of big F.", 'start': 523.261, 'duration': 2.122}], 'summary': 'The fourier transform of big f, denoted as big f hat, is a function of omega and equals the integral of big f of t e to the minus i, omega t dt.', 'duration': 26.141, 'max_score': 499.242, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/7UvtU75NXTg/pics/7UvtU75NXTg499242.jpg'}, {'end': 601.097, 'src': 'embed', 'start': 572.967, 'weight': 3, 'content': [{'end': 576.37, 'text': "So I've changed my bounds of integration because of my Heaviside function.", 'start': 572.967, 'duration': 3.403}, {'end': 580.295, 'text': "So instead of negative infinity to infinity, I'm doing 0 to infinity.", 'start': 576.611, 'duration': 3.684}, {'end': 587.262, 'text': "And then what I'm going to do is I'm going to group these exponentials here.", 'start': 582.537, 'duration': 4.725}, {'end': 601.097, 'text': "and i'm going to say that this equals integral from 0 to infinity little f, of t e to the minus gamma plus i, omega t dt.", 'start': 588.223, 'duration': 12.874}], 'summary': 'Changed integration bounds from -infinity to 0 to infinity, grouping exponentials.', 'duration': 28.13, 'max_score': 572.967, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/7UvtU75NXTg/pics/7UvtU75NXTg572967.jpg'}, {'end': 643.23, 'src': 'heatmap', 'start': 616.116, 'weight': 0.713, 'content': [{'end': 624.262, 'text': 'And so this equals integral zero to infinity f of t e to the st dt.', 'start': 616.116, 'duration': 8.146}, {'end': 627.083, 'text': 'And that is my Laplace transform.', 'start': 625.142, 'duration': 1.941}, {'end': 632.105, 'text': 'The Fourier transform of big F is my Laplace transform of little f.', 'start': 627.143, 'duration': 4.962}, {'end': 634.206, 'text': "That's the definition of the Laplace transform.", 'start': 632.105, 'duration': 2.101}, {'end': 643.23, 'text': "It's the Laplace transform, the Laplace transform is the Fourier transform of a one-sided weighted function f.", 'start': 634.666, 'duration': 8.564}], 'summary': 'Laplace transform is the fourier transform of a one-sided weighted function f.', 'duration': 27.114, 'max_score': 616.116, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/7UvtU75NXTg/pics/7UvtU75NXTg616116.jpg'}, {'end': 986.415, 'src': 'embed', 'start': 950.746, 'weight': 0, 'content': [{'end': 960.693, 'text': 'and it is extremely useful because lots of the solutions of PDEs and ODEs and control theory look more like this and this and this than our nicely behaved functions,', 'start': 950.746, 'duration': 9.947}, {'end': 962.194, 'text': 'where we can easily Fourier transform.', 'start': 960.693, 'duration': 1.501}, {'end': 967.358, 'text': "So in the next lecture, what I'm going to do is I'm going to walk you through some of the properties of the Laplace transform.", 'start': 962.914, 'duration': 4.444}, {'end': 970.961, 'text': 'It inherits most of the same properties as the Fourier transform.', 'start': 967.438, 'duration': 3.523}, {'end': 974.244, 'text': 'For example, how you transform derivatives or convolutions.', 'start': 971.342, 'duration': 2.902}, {'end': 982.091, 'text': "And we're going to use those properties of the Laplace transform to simplify our PDEs to ODEs, our ODEs to algebraic equations.", 'start': 974.805, 'duration': 7.286}, {'end': 985.174, 'text': "And we're also going to use this a lot in the control theory boot camp.", 'start': 982.471, 'duration': 2.703}, {'end': 985.874, 'text': 'All right.', 'start': 985.634, 'duration': 0.24}, {'end': 986.415, 'text': 'Thank you.', 'start': 986.114, 'duration': 0.301}], 'summary': 'Laplace transform simplifies pdes to odes, used in control theory boot camp.', 'duration': 35.669, 'max_score': 950.746, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/7UvtU75NXTg/pics/7UvtU75NXTg950746.jpg'}], 'start': 499.242, 'title': 'Fourier and laplace transforms', 'summary': 'Covers the fourier transform, demonstrating the use of heaviside function to change bounds of integration, and discusses laplace transform properties, providing a method to solve pdes and odes and its applications in control theory.', 'chapters': [{'end': 601.097, 'start': 499.242, 'title': 'Fourier transform and bounds of integration', 'summary': 'Discusses the fourier transform of a function, demonstrating the use of heaviside function to change the bounds of integration, resulting in a transformed function from -∞ to ∞ to 0 to ∞, with the grouped exponentials in the integral.', 'duration': 101.855, 'highlights': ['The use of the Fourier transform to transform a function is explained, emphasizing the integral from -∞ to ∞, and the resulting transformed function denoted as big F hat.', 'The utilization of the Heaviside function to modify the bounds of integration, changing the integral from -∞ to ∞ to 0 to ∞, allowing for the simplification of the expression.', 'The grouping of exponentials in the integral is demonstrated, showcasing the manipulation of the function using the combined exponentials of e to the -γt and e to the -iωt.']}, {'end': 986.415, 'start': 601.097, 'title': 'Laplace transform properties', 'summary': 'Discusses the definition and properties of the laplace transform, which is a generalized fourier transform for badly behaved functions, providing a method to solve pdes and odes and its applications in control theory.', 'duration': 385.318, 'highlights': ['The Laplace transform is a one-sided, weighted Fourier transform for badly behaved functions. It involves multiplying badly behaved functions by a stable exponential and a Heaviside function, then Fourier transforming the product.', 'The Laplace transform is extremely useful for solving PDEs and ODEs in control theory. It allows simplifying PDEs to ODEs, ODEs to algebraic equations, and finds applications in control theory.', 'The Laplace transform inherits most properties of the Fourier transform, like transforming derivatives or convolutions. It shares similar properties with the Fourier transform, facilitating the simplification of mathematical equations.']}], 'duration': 487.173, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/7UvtU75NXTg/pics/7UvtU75NXTg499242.jpg', 'highlights': ['The Laplace transform is extremely useful for solving PDEs and ODEs in control theory. It allows simplifying PDEs to ODEs, ODEs to algebraic equations, and finds applications in control theory.', 'The Laplace transform inherits most properties of the Fourier transform, like transforming derivatives or convolutions. It shares similar properties with the Fourier transform, facilitating the simplification of mathematical equations.', 'The use of the Fourier transform to transform a function is explained, emphasizing the integral from -∞ to ∞, and the resulting transformed function denoted as big F hat.', 'The utilization of the Heaviside function to modify the bounds of integration, changing the integral from -∞ to ∞ to 0 to ∞, allowing for the simplification of the expression.', 'The grouping of exponentials in the integral is demonstrated, showcasing the manipulation of the function using the combined exponentials of e to the -γt and e to the -iωt.']}], 'highlights': ['The Laplace transform is extensively useful in control theory, making it a powerful tool in engineering and mathematics.', 'The Laplace transform is extremely useful for solving PDEs and ODEs in control theory. It allows simplifying PDEs to ODEs, ODEs to algebraic equations, and finds applications in control theory.', 'Applying the Laplace transform can simplify solving systems by subtracting advanced math, making the process much simpler.', 'Under certain conditions, the Laplace transform can transform a PDE into an ODE, simplifying the solution techniques from college to high school level.', 'The Laplace transform is a generalized version of the Fourier transform, simplifying the solution of systems and converting PDEs to ODEs.', 'The Laplace transform inherits most properties of the Fourier transform, like transforming derivatives or convolutions. It shares similar properties with the Fourier transform, facilitating the simplification of mathematical equations.', 'Laplace was one of the first researchers to realize the importance of looking at real-world data through a probabilistic lens.', 'The solution involves multiplying a function by a decaying exponential function so that f(t) * e^(-gamma*t) goes to 0 as t goes to positive infinity.', 'The use of stable exponential and Heaviside function to stabilize a badly behaved function, allowing its Fourier transformation.', 'Defining big function big F of t as little f times e to the minus gamma t times heavy side of t.', 'The Laplace transform of little f is the Fourier transform of big F.', 'This approach is used to make the badly behaved function approach zero as t goes to positive infinity.', 'Laplace transform as a weighted, one-sided Fourier transform for nastier functions, addressing the limitations of Fourier transform', 'Demonstration of the ability of Fourier transform to handle well-behaved functions that decay to zero at plus and minus infinity', 'Explanation of the limitations of Fourier transform in dealing with nastier functions such as e to the lambda t and the Heaviside function', 'Introduction of the concept of using a window function to multiply with trigonometric functions for Fourier transformation, albeit with complexities', 'The use of the Fourier transform to transform a function is explained, emphasizing the integral from -∞ to ∞, and the resulting transformed function denoted as big F hat.', 'The utilization of the Heaviside function to modify the bounds of integration, changing the integral from -∞ to ∞ to 0 to ∞, allowing for the simplification of the expression.', 'The grouping of exponentials in the integral is demonstrated, showcasing the manipulation of the function using the combined exponentials of e to the -γt and e to the -iωt.']}